Solve the Rational Equation for x square root of x-5 = square root of x+4

$\sqrt{x} - 5 = \sqrt{x + 4}$

Step 1

Add $5$ to both sides of the equation.

$\sqrt{x} = \sqrt{x + 4} + 5$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(\sqrt{x + 4} + 5)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{x + 4} + 5)}^{2}$

Simplify the left side.

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 4} + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 4} + 5)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{x + 4} + 5)}^{2}$

Simplify.

$x = {(\sqrt{x + 4} + 5)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 4} + 5)}^{2}$ .

Rewrite ${(\sqrt{x + 4} + 5)}^{2}$ as $(\sqrt{x + 4} + 5) (\sqrt{x + 4} + 5)$ .

$x = (\sqrt{x + 4} + 5) (\sqrt{x + 4} + 5)$

Expand $(\sqrt{x + 4} + 5) (\sqrt{x + 4} + 5)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x + 4} (\sqrt{x + 4} + 5) + 5 (\sqrt{x + 4} + 5)$

Apply the distributive property.

$x = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot 5 + 5 (\sqrt{x + 4} + 5)$

Apply the distributive property.

$x = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

Raise $\sqrt{x + 4}$ to the power of $1$ .

$x = {\sqrt{x + 4}}^{1} \sqrt{x + 4} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Raise $\sqrt{x + 4}$ to the power of $1$ .

$x = {\sqrt{x + 4}}^{1} {\sqrt{x + 4}}^{1} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = {\sqrt{x + 4}}^{1 + 1} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Add $1$ and $1$ .

$x = {\sqrt{x + 4}}^{2} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$x = {({(x + 4)}^{\frac{1}{2}})}^{2} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = {(x + 4)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Combine $\frac{1}{2}$ and $2$ .

$x = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Rewrite the expression.

$x = {(x + 4)}^{1} + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Simplify.

$x = x + 4 + \sqrt{x + 4} \cdot 5 + 5 \sqrt{x + 4} + 5 \cdot 5$

Move $5$ to the left of $\sqrt{x + 4}$ .

$x = x + 4 + 5 \cdot \sqrt{x + 4} + 5 \sqrt{x + 4} + 5 \cdot 5$

Multiply $5$ by $5$ .

$x = x + 4 + 5 \sqrt{x + 4} + 5 \sqrt{x + 4} + 25$

Add $4$ and $25$ .

$x = x + 29 + 5 \sqrt{x + 4} + 5 \sqrt{x + 4}$

Add $5 \sqrt{x + 4}$ and $5 \sqrt{x + 4}$ .

$x = x + 29 + 10 \sqrt{x + 4}$

Step 4

Solve for $10 \sqrt{x + 4}$ .

Rewrite the equation as $x + 29 + 10 \sqrt{x + 4} = x$ .

$x + 29 + 10 \sqrt{x + 4} = x$

Move all terms not containing $10 \sqrt{x + 4}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$29 + 10 \sqrt{x + 4} = x - x$

Subtract $29$ from both sides of the equation.

$10 \sqrt{x + 4} = x - x - 29$

Subtract $x$ from $x$ .

$10 \sqrt{x + 4} = 0 - 29$

Subtract $29$ from $0$ .

$10 \sqrt{x + 4} = - 29$

Step 5

To remove the radical on the left side of the equation, square both sides of the equation.

${(10 \sqrt{x + 4})}^{2} = {(- 29)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

${(10 {(x + 4)}^{\frac{1}{2}})}^{2} = {(- 29)}^{2}$

Simplify the left side.

Simplify ${(10 {(x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $10 {(x + 4)}^{\frac{1}{2}}$ .

$10^{2} {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 29)}^{2}$

Raise $10$ to the power of $2$ .

$100 {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 29)}^{2}$

Multiply the exponents in ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$100 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 29)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 29)}^{2}$

Rewrite the expression.

$100 {(x + 4)}^{1} = {(- 29)}^{2}$

Simplify.

$100 (x + 4) = {(- 29)}^{2}$

Apply the distributive property.

$100 x + 100 \cdot 4 = {(- 29)}^{2}$

Multiply $100$ by $4$ .

$100 x + 400 = {(- 29)}^{2}$

Simplify the right side.

Raise $- 29$ to the power of $2$ .

$100 x + 400 = 841$

Step 7

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $400$ from both sides of the equation.

$100 x = 841 - 400$

Subtract $400$ from $841$ .

$100 x = 441$

Divide each term in $100 x = 441$ by $100$ and simplify.

Divide each term in $100 x = 441$ by $100$ .

$\frac{100 x}{100} = \frac{441}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{441}{100}$

Divide $x$ by $1$ .

$x = \frac{441}{100}$

Step 8

Exclude the solutions that do not make $\sqrt{x} - 5 = \sqrt{x + 4}$ true.

$No solution$

Solve the Rational Equation for x square root of x-5 = square root of x+4

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